Problem: $\overline{AC} = 24$ $\overline{BC} = {?}$ $A$ $C$ $B$ $24$ $?$ $ \sin( \angle BAC ) = \dfrac{5}{13}, \cos( \angle BAC ) = \dfrac{12}{13}, \tan( \angle BAC ) = \dfrac{5}{12}$
Solution: $\overline{BC}$ is the opposite to $\angle BAC$ $\overline{AC}$ is adjacent to $\angle BAC$ SOH CAH TOA We know the adjacent side and need to solve for the opposite side so we can use the tan function (TOA) $ \tan( \angle BAC ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\overline{BC}}{\overline{AC}}= \frac{\overline{BC}}{24} $ Since we have already been given $\tan( \angle BAC )$ , we can set up a proportion to find $\overline{BC}$ $ \tan( \angle BAC ) = \dfrac{5}{12} = \frac{\overline{BC}}{24}$ Simplify. $\overline{BC} = 10$